(with some illustrations borrowed from Kevin Gurney's notes, and
some descriptions borrowed from "Neural networks and physical systems
with emergent collective computational abilities" by John Hopfield)
The purpose of a Hopfield net is to store 1 or more patterns and to recall the full patterns based on partial input. For example, consider the problem of optical character recognition. The task is to scan an input text and extract the characters out and put them in a text file in ASCII form. Okay, so what happens if you spilled coffee on the text that you want to scan? Now some of the characters are not quite as well defined, though they're mostly closer to the original characters than any other character:
So here's the way a Hopfield network would work. You map it out so that each pixel is one node in the network. You train it (or just assign the weights) to recognize each of the 26 characters of the alphabet, in both upper and lower case (that's 52 patterns). Now if your scan gives you a pattern like something on the right of the above illustration, you input it to the Hopfield network, and it chugs away for a few iterations, and eventually reproduces the pattern on the left, a perfect "T". Note that this could work with higherlevel chunks; for example, it could have an array of pixels to represent the whole word. It could also be used for something more complex like sound or facial images. The problem is, the more complex the things being recalled, the more pixels you need, and as you will see, if you have N pixels, you'll be dealing with N^{2} weights, so the problem is very computationally expensive (and thus slow).

All the nodes in a Hopfield network are both inputs and outputs, and they are fully interconnected. That is, each node is an input to every other node in the network. You can think of the links from each node to itself as being a link with a weight of 0. Here's a picture of a 3node Hopfield network: 
In the previous neural models you've seen, the processing (ignoring training) only goes in one direction: you start from the input nodes, do a sum & threshold of those values to get the outputs of the first layer, and possibly pass those values to a second layer of summing & thresholding, but nothing gets passed back from layer 2 to layer 1 or even passed between the nodes in layer 1: 
In a Hopfield network, all the nodes are inputs to each other, and
they're also outputs. As I stated above, how it works in
computation is that you put a distorted pattern onto the nodes of
the network, iterate a bunch of times, and eventually it arrives at
one of the patterns we trained it to know and stays there. So, what
you need to know to make it work are:

There are ways of training a Hopfield network in the sense that you're used to, and you can read about those in Chapter 6 of Kevin Gurney's notes. In the original article where Hopfield described the net, he provided a formula with which you can calculate the values of the weights without any training. (I'll try to use notation that you're familiar with, like using W_{ij} to represent the weight from node i to node j)Suppose we wish to store the set of states V^{s}, s = 1, ..., n. We use the storage prescription: 
W_{ij} = 
(2V^{s}_{i}  1)(2V^{s}_{j}  1) 
[1] 
but with W_{ij} = 0.Note that if you only have one pattern, this equation deteriorates to: 
W_{ij} = 
(2V_{i}  1)(2V_{j}  1) 
[2] 
For example, say we have a 5 node Hopfield network and we want it to recognize the pattern (0 1 1 0 1). Since there are 5 nodes, we need a matrix of 5 x 5 weights, where the weights from a node back to itself are 0. The weight matrix will look like this: 
0  W_{12}  W_{13}  W_{14}  W_{15}  
W_{21}  0  W_{23}  W_{24}  W_{25}  
W_{31}  W_{32}  0  W_{34}  W_{35}  
W_{41}  W_{42}  W_{43}  0  W_{45}  
W_{51}  W_{52}  W_{53}  W_{54}  0 
One thing you might notice from equation [2] is that if you switch the i and j indexes, you get the same result. That is,
W_{ij} = (2V_{i}  1)(2V_{j}  1) = (2V_{j}  1)(2V_{i}  1) = W_{ji}Since the weights are symmetric, we only have to calculate the upper diagonal of weights, and then we can copy each weight to its inverse weight. In this case, V is the vector (0 1 1 0 1), so V_{1} = 0, V_{2} = 1, V_{3} = 1, V_{4} = 0, and V_{5} = 1. Thus the computation of the weights is as follows: 
W_{12} = (2V_{1}  1)(2V_{2}  1) = (0 
1)(2  1) = (1)(1) = 1

So now our weight matrix looks like this: 
0  1  1  1  1  
W_{21}  0  1  1  1  
W_{31}  W_{32}  0  1  1  
W_{41}  W_{42}  W_{43}  0  1  
W_{51}  W_{52}  W_{53}  W_{54}  0 
By reflecting about the diagonal, we get the full weight matrix: 
0  1  1  1  1  
1  0  1  1  1  
1  1  0  1  1  
1  1  1  0  1  
1  1  1  1  0 
For completeness' sake, you'll remember that the original formula was set up to allow you to have n patterns. So let's consider the case where we want our 5 node Hopfield net to store both the pattern V^{1} = (0 1 1 0 1) and another pattern V^{2} = (1 0 1 0 1). One way you could go about calculating the weights is on a weight by weight basis. For example, W_{12} could be calculated as:

W_{12} = 
(2V^{s}_{1}  1)(2V^{s}_{2}  1)  
= 
(2V^{1}_{1}  1)(2V^{1}_{2}  1) + (2V^{2}_{1}  1)(2V^{2}_{2}  1)  
= 
(2*0  1)(2*1  1) + (2*1  1)(2*0  1)  
= 
(0  1)(2  1) + (2  1)(0  1)  
= 
(1)(1) + (1)(1)  
= 
1 + 1  
= 
2 
You could go through each weight like this and calculate the new weight matrix. This is likely the way you would do it by computer, but I find if I'm calculating them by hand, it's easier to focus on one pattern at a time and then sum the resulting matrices. To do this, you would calculate the matrix for the first pattern (which we did above), then calculate the value for the second matrix and finally add the two matrices together. Here's the weight matrix for the pattern (1 0 1 0 1): 
0  1  1  1  1  
1  0  1  1  1  
1  1  0  1  1  
1  1  1  0  1  
1  1  1  1  0 
Now add this to the previous weight matrix and we get: 
0  2  0  0  0  
2  0  0  0  0  
0  0  0  2  2  
0  0  2  0  2  
0  0  2  2  0 
So now we have a weight matrix for a 5 node Hopfield network that's meant to recognize the patterns (0 1 1 0 1) and (1 0 1 0 1). One thing you might notice is that these patterns only differ by 2 bits. As you might imagine, patterns this close together might be difficult to tell apart. By analogy, you might have trouble discriminating a lower case "c" from "e" or an upper case "O" from "Q" if they were mangled badly enough. Let's start from the pattern (1 1 1 1 1), which only differs from each of these patterns by 2 bits, and see what happens.Updating a node in a Hopfield network is very much like updating a perceptron. If you are updating node 3 of a Hopfield network, then you can think of that as the perceptron, and the values of all the other nodes as input values, and the weights from those nodes to node 3 as the weights. In other words, first you do a weighted sum of the inputs from the other nodes, then if that value is greater than or equal to 0, you output 1. Otherwise, you output 0. In formula form: 
V_{i}in = 
W_{ji}V_{j}  
V_{i} > 1 if V_{i}in >= 0  
else V_{i} > 0 
So assuming we have the final weight matrix from the previous section, and we start from the state (1 1 1 1 1), for the 3rd node, we have: 
V_{3}in = 
W_{j3}V_{j}  
= 
W_{13}V_{1} + W_{23}V_{2} + W_{43}V_{4} + W_{53}V_{5}  
= 
0*1 + 0*1 + 2*1 + 2*1  
= 
0  
since 0 >= 0,  
V_{3} = 1 
In this case, the value of V_{3} doesn't change. It's worth noticing that since the weight from node 3 to itself is 0, we could have just calculated the dot product of the 3rd column out of the weight matrix and the current state to calculate the weighted sum: 
V_{3}in = (0 0 0 2 2) * (1 1 1 1 1) = 2 + 2 = 0 
You might have noticed by now that sequencing the updates of the nodes in a Hopfield network is somewhat tricky. How can you update a neuron if the values of its inputs are changing? Well, there are two approaches. The first is synchonous updating, which means all the nodes get updated at the same time, based on the existing state (i.e. not on the values the nodes are changing to). To update the nodes in this method, you can just multiply the weight matrix by the vector of the current state.This isn't very realistic in a neural sense, as neurons don't all update at the same rate. They have varying propagation delays, varying firing times, etc., so a more realistic assumption would be to update them in random order. This was the method described by Hopfield, in fact. You randomly select a neuron, and update it. Then you randomly select another neuron and update it. You keep doing this until the system is in a stable state (which we'll talk about later). In practice, people code Hopfield nets in a semirandom order. They update all of the nodes in one step, but within that step they are updated in random order. So it might go 3, 2, 1, 5, 4, 2, 3, 1, 5, 4, etc. This is just to avoid a bad pseudorandom generator from favoring one of the nodes, which could happen if it was purely random: 3, 2, 1, 2, 2, 2, 5, 1, 2, 2, 4, 2, 1, etc. 
The main reason to want to cycle through all the nodes each step is that it's the only way you can tell when to stop. Basically, if you go through all the nodes and none of them changes, you can stop. If you're updating them in a fixed sequence (e.g. 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, etc.), this means that if you go through 5 consecutive neurons without changing any of their values, then you're at an attractor so you can stop. 
Now let's finish the example I started. In other words, given the
weight matrix for a 5 node network with (0 1 1 0 1) and
(1 0 1 0 1) as attractors, start at the state (1 1 1 1 1) and see
where it goes. To keep it simple, I'm going to update the nodes
in the fixed order 3, 1, 5, 2, 4, 3, 1, 5, 2, 4, etc. I already
did the first update of node 3, and it didn't change, so
continuing:
V_{1}in = (0 2 0 0 0) . (1 1 1 1 1) = 2 since 2 < 0, V_{1} = 0 (it changed) V_{5}in = (0 0 2 2 0) . (0 1 1 1 1) = 0 since 0 >= 0, V_{5} = 1 (it didn't change) V_{2}in = (2 0 0 0 0) . (0 1 1 1 1) = 0 since 0 >= 0, V_{2} = 1 (it didn't change) V_{4}in = (0 0 2 0 2) . (0 1 1 1 1) = 4 since 4 < 0, V_{4} = 0 (it changed) V_{3}in = (0 0 0 2 2) . (0 1 1 0 1) = 2 since 2 >= 0, V_{3} = 1 (it didn't change) V_{1}in = (0 2 0 0 0) . (0 1 1 0 1) = 2 since 2 < 0, V_{1} = 0 (it didn't change) V_{5}in = (0 0 2 2 0) . (0 1 1 0 1) = 2 since 2 >= 0, V_{5} = 1 (it didn't change) V_{2}in = (2 0 0 0 0) . (0 1 1 0 1) = 0 since 0 >= 0, V_{2} = 1 (it didn't change) V_{4}in = (0 0 2 0 2) . (0 1 1 0 1) = 4 since 4 < 0, V_{4} = 0 (it didn't change) 